Run actions with a custom button
Too add custom buttons to the UI here in 3D view / Toolpanel you could something like this:

import bpy

class DissolvePanel(bpy.types.Panel):
    bl_label = "Dissolve Panel"
    bl_idname = "OBJECT_PT_hello"
    bl_space_type = 'VIEW_3D'
    bl_region_type = 'TOOLS'

    def draw(self, context):
        layout = self.layout
        col = layout.column(align=True)
        row = col.row(align=True)
        row.operator("mesh.dissolve_edges", text = "Dissolve Edges")
        row.operator("mesh.dissolve_verts", text = "Dissolve Verts")
        # invoke custom operator
        # row.operator("object.simple_operator" , text = "All")

def register():
    bpy.utils.register_class(DissolvePanel)

def unregister():
    bpy.utils.unregister_class(DissolvePanel)

if __name__ == "__main__":
    register()


To run dissolve edges followed by dissolve vertices you would need to define your own operator, to enable the All button uncomment this line in the above script:

# row.operator("object.simple_operator" , text = "All"):


Custom operator (from editor's templates)

import bpy


def main(context):
    bpy.ops.mesh.dissolve_edges()
    bpy.ops.mesh.dissolve_verts()

class SimpleOperator(bpy.types.Operator):
    """Tooltip"""
    bl_idname = "object.simple_operator"
    bl_label = "Simple Object Operator"

    @classmethod
    def poll(cls, context):
        return context.active_object is not None

    def execute(self, context):
        main(context)
        return {'FINISHED'}

def register():
    bpy.utils.register_class(SimpleOperator)

def unregister():
    bpy.utils.unregister_class(SimpleOperator)

if __name__ == "__main__":
    register()
    # test call
    bpy.ops.object.simple_operator()